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When this study was performed, A. A. Kokhanovsky was with the Earth Observation Research Center, National Space Development Agency of Japan, 1-9-9, Roppongi, Minato-ku, Tokyo 106, Japan. He is now with the Institute of Physics, Academy of Sciences of Belarus, 70 Skarina Prospekt, Minsk 220070,
Belarus.

Abstract

We obtain and analyze simple analytical formulas for asymmetry parameters and absorption cross sections of large, nonspherical particles. The formulas are based on the asymptotic properties of these characteristics at strong and weak absorption of radiation inside particles. The absorption cross section depends on parameter ϕ, which determines the value of the light-absorption cross section for weakly absorbing particles. It is larger for nonspherical scatterers. The asymmetry parameter depends on two parameters. The first is the asymmetry parameter g0 of a nonspherical, transparent particle with the same shape as an absorbing one. The second parameter, β, determines the strength of the influence of light absorption on the value of the asymmetry parameter. Parameter β is larger for nonspherical particles. One can find these three parameters (ϕ, g0, and β) using a ray-tracing code (RTC) for nonabsorbing and weakly absorbing particles. The RTC can then be used to check the accuracy of the equations at any absorption for hexagonal cylinders and spheroids. It is found that the error of computing the absorption cross section and 1 − g (g is the asymmetry parameter) is less than 20% at the refractive index of particles n = 1.333. Values for asymmetry parameters of large, nonabsorbing, spheroidal particles with different aspect ratios are tabulated for the first time to our knowledge. They do not depend on the size of particles and can serve as an independent check of the accuracy of T-matrix codes for large parameters.

References

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aa1, a2, and a3, are semiaxes of a spheroid, ξ = a3/a1, for an oblate spheroid δ = (1 − ξ2)1/2, and for a prolate spheroid δ = (1/ξ)(ξ2 − 1)1/2. L is the length of a hexagonal cylinder, and l is the side of a hexagonal cylinder’s cross section.

Table 3

Value of g0 for Randomly Oriented, Spheroidal Particles at Different Values of Shape Parameter ξ and the Real Part of Refractive Index n

n/ξ

0.3

0.5

0.7

1.0

1.5

2.0

3.5

∞

1.1

0.9582

0.9583

0.9764

0.9731

0.9652

0.9642

0.9711

0.9817

1.2

0.9123

0.8976

0.9127

0.9341

0.9154

0.9158

0.9413

0.9551

1.333

0.8774

0.8129

0.8428

0.8843

0.8510

0.8556

0.9085

0.9201

1.4

0.8590

0.7718

0.8041

0.8613

0.8208

0.8298

0.8911

0.9029

1.5

0.8257

0.7135

0.7522

0.8299

0.7747

0.7965

0.8624

0.8795

1.6

0.7873

0.6677

0.7065

0.8015

0.7322

0.7685

0.8320

0.8577

1.7

0.7518

0.6407

0.6665

0.7759

0.6934

0.7479

0.8020

0.8360

Table 4

Values of β(n, ξ) for Randomly Oriented, Spheroidal Particles at Different Values of Shape Parameter ξ and the Real Part of Refractive Index n

n/ξ

0.3

0.5

0.7

1.0

1.5

2.0

3.5

1.1

2.49

1.32

0.90

0.47

0.86

1.04

1.43

1.2

2.21

1.59

0.83

0.54

1.05

1.10

1.37

1.333

3.09

1.59

1.16

0.76

1.09

1.27

1.40

1.4

3.21

1.66

1.26

0.82

1.13

1.32

1.48

1.5

3.53

1.73

1.29

0.83

1.18

1.32

1.65

1.6

3.70

1.78

1.37

0.86

1.23

1.35

1.83

1.7

3.83

1.84

1.43

0.89

1.28

1.40

1.98

Table 5

Values of ϕ(n, ξ) for Randomly Oriented, Spheroidal Particles at Different Values of Shape Parameter ξ and the Real Part of Refractive Index n

n/ξ

0.3

0.5

0.7

1.0

1.5

2.0

3.5

1.1

2.46

1.66

1.32

1.11

1.14

1.18

1.24

1.2

2.78

1.84

1.45

1.18

1.25

1.29

1.37

1.333

3.16

2.09

1.60

1.24

1.36

1.43

1.52

1.4

3.35

2.21

1.67

1.26

1.42

1.49

1.59

1.5

3.64

2.40

1.78

1.29

1.50

1.58

1.70

1.6

3.93

2.59

1.88

1.31

1.57

1.68

1.80

1.7

4.21

2.76

2.00

1.33

1.64

1.76

1.89

Table 6

Values of ψ, β, and g0 for Hexagonal Cylinders at n = 1.333 and Different Values of the Ratio ν = L/la

ν

ψ

β

g0

0.2

1.9

4.8

0.9031

0.4

1.8

1.7

0.8607

1

1.6

1.1

0.7847

2

1.6

0.9

0.7601

4

1.8

0.7

0.7971

10

1.9

1.5

0.8442

20

1.9

1.5

0.8665

a Here L is the length of a hexagonal cylinder and l is the side of a hexagon’s cross section.

aa1, a2, and a3, are semiaxes of a spheroid, ξ = a3/a1, for an oblate spheroid δ = (1 − ξ2)1/2, and for a prolate spheroid δ = (1/ξ)(ξ2 − 1)1/2. L is the length of a hexagonal cylinder, and l is the side of a hexagonal cylinder’s cross section.

Table 3

Value of g0 for Randomly Oriented, Spheroidal Particles at Different Values of Shape Parameter ξ and the Real Part of Refractive Index n

n/ξ

0.3

0.5

0.7

1.0

1.5

2.0

3.5

∞

1.1

0.9582

0.9583

0.9764

0.9731

0.9652

0.9642

0.9711

0.9817

1.2

0.9123

0.8976

0.9127

0.9341

0.9154

0.9158

0.9413

0.9551

1.333

0.8774

0.8129

0.8428

0.8843

0.8510

0.8556

0.9085

0.9201

1.4

0.8590

0.7718

0.8041

0.8613

0.8208

0.8298

0.8911

0.9029

1.5

0.8257

0.7135

0.7522

0.8299

0.7747

0.7965

0.8624

0.8795

1.6

0.7873

0.6677

0.7065

0.8015

0.7322

0.7685

0.8320

0.8577

1.7

0.7518

0.6407

0.6665

0.7759

0.6934

0.7479

0.8020

0.8360

Table 4

Values of β(n, ξ) for Randomly Oriented, Spheroidal Particles at Different Values of Shape Parameter ξ and the Real Part of Refractive Index n

n/ξ

0.3

0.5

0.7

1.0

1.5

2.0

3.5

1.1

2.49

1.32

0.90

0.47

0.86

1.04

1.43

1.2

2.21

1.59

0.83

0.54

1.05

1.10

1.37

1.333

3.09

1.59

1.16

0.76

1.09

1.27

1.40

1.4

3.21

1.66

1.26

0.82

1.13

1.32

1.48

1.5

3.53

1.73

1.29

0.83

1.18

1.32

1.65

1.6

3.70

1.78

1.37

0.86

1.23

1.35

1.83

1.7

3.83

1.84

1.43

0.89

1.28

1.40

1.98

Table 5

Values of ϕ(n, ξ) for Randomly Oriented, Spheroidal Particles at Different Values of Shape Parameter ξ and the Real Part of Refractive Index n

n/ξ

0.3

0.5

0.7

1.0

1.5

2.0

3.5

1.1

2.46

1.66

1.32

1.11

1.14

1.18

1.24

1.2

2.78

1.84

1.45

1.18

1.25

1.29

1.37

1.333

3.16

2.09

1.60

1.24

1.36

1.43

1.52

1.4

3.35

2.21

1.67

1.26

1.42

1.49

1.59

1.5

3.64

2.40

1.78

1.29

1.50

1.58

1.70

1.6

3.93

2.59

1.88

1.31

1.57

1.68

1.80

1.7

4.21

2.76

2.00

1.33

1.64

1.76

1.89

Table 6

Values of ψ, β, and g0 for Hexagonal Cylinders at n = 1.333 and Different Values of the Ratio ν = L/la

ν

ψ

β

g0

0.2

1.9

4.8

0.9031

0.4

1.8

1.7

0.8607

1

1.6

1.1

0.7847

2

1.6

0.9

0.7601

4

1.8

0.7

0.7971

10

1.9

1.5

0.8442

20

1.9

1.5

0.8665

a Here L is the length of a hexagonal cylinder and l is the side of a hexagon’s cross section.